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Probing the sub-dominant orders in the large-N limit of colored tensor models

Wojciech Kamiński, Daniele Oriti, Ryan, J.

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

The definition of a double-scaling limit represents an important goal in the development of tensor models.We take the first steps towards this goal by extracting and analysing the next-to-leading order contributions, in the 1/N expansion, for the iid tensor models.We show that the radius of convergence of the nlo series coincides with that of the leading order melonic sector.Meanwhile, the value of the susceptibility exponent, γnlo = 3/2, signals a departure from the leading order behaviour.Both pieces of information provide clues for a non-trivial double-scaling limit, for which we put forward some precise conjecture.

Related papers

Towards a double scaling limit for tensor models: probing sub-dominant orders

Wojciech Kamiński, Daniele Oriti, James P. Ryan

The definition of a double scaling limit represents an important goal in the development of tensor models. We take the first steps towards this goal by extracting and analysing the next-to-leading order contributions, in the 1/N expansion, for the colored tensor models. We show that the radius of convergence of the nlo series coincides with that of the leading order melonic sector. Meanwhile, the value of the susceptibility exponent, γ nlo = 3/2, signals a departure from the leading order behavior. Both pieces of information provide clues for a non-trivial double scaling limit, for which we put forward some precise conjecture.

arXiv:1304.6934•New Journal of Physics•Jun 19, 2014•59 citations

The double scaling limit of the multi-orientable tensor model

Razvan Gurău, Adrian Tanasă, Donald R. Youmans

In this paper we study the double scaling limit of the multi-orientable tensor model. We prove that, contrary to the case of matrix models but similarly to the case of invariant tensor models, the double scaling series are convergent. We resum the double scaling series of the two-point function and of the leading singular part of the four-point function. We discuss the behavior of the leading singular part of arbitrary correlation functions. We show that the contribution of the four-point function and of all the higher point functions are enhanced in the double scaling limit. We finally show that all the correlation functions exhibit a singularity at the same critical value of the double scaling parameter which, combined with the convergence of the double scaling series, suggest the existence of a triple scaling limit.

arXiv:1505.00586•Europhysics Letters (EPL)•Jul 1, 2015•52 citations

Another proof of the $1/N$ expansion of the rank three tensor model with tetrahedral interaction

Valentin Bonzom

The rank three tensor model with tetrahedral interaction was shown by Carrozza and Tanasa to admit a $1/N$ expansion, dominated by melonic diagrams, and double tadpoles decorated with melons at next-to-leading order. This model has generated a renewed interest in tensor models because it has the same large $N$ limit as the SYK model. In contrast with matrix models, there is no method which would be able to prove the existence of $1/N$ expansions in arbitrary tensor models. The method used by Carrozza and Tanasa proves the existence of the $1/N$ expansion using two-dimensional topology, before identifying the leading order and next-to-leading graphs. However, another method was required for complex, rank three tensor models with planar interactions, which is based on flips. The latter are moves which cut two propagators of Feynman graphs and reglue them differently. They allow transforming graphs while tracking their orders in the $1/N$ expansion. Here we use this method to re-prove the results of Carrozza and Tanasa, thereby proving the existence of the $1/N$ expansion, the melonic dominance at leading order and the melon-decorated double tadpoles at next-to-leading order, all in one go.

arXiv:1912.11104•Dec 23, 2019•2 citations

Random tensor models—theU(N)D-invariant model

Adrian Tanasă

Abstract In the first section we give a briefly presentation of the U(N)D-invariant tensor models (N being again the size of the tensor, and D being the dimension). The next section is then dedicated to the analysis of the Dyson–Schwinger equations (DSE) in the large N limit. These results are essential to implement the double scaling limit mechanism of the DSEs, which is done in the third section. The main result of this chapter is the doubly-scaled 2-point function for a model with generic melonic interactions. However, several assumptions on the large N scaling of cumulants are made along the way. They are proved using various combinatorial methods.

•Apr 16, 2021•0 citations

Notes on tensor models and tensor field theories

Razvan Gurău

Tensor models and tensor field theories admit a 1/N expansion and a melonic large N limit which is simpler than the planar limit of random matrices and richer than the large N limit of vector models. They provide examples of analytically tractable but non-trivial strongly coupled quantum field theories and lead to a new class of conformal field theories. We present a compact introduction to the topic, covering both some of the classical results in the field, like the details of the 1/N expansion, as well as recent developments. These notes are loosely bases on four lectures given at the Journées de physique mathématique Lyon 2019: Random tensors and SYK models.

arXiv:1907.03531•Annales de l’Institut Henri Poincaré D Combinatorics Physics and their Interactions•Apr 11, 2022•40 citations

The double scaling limit

Razvan Gurău

Abstract This chapter presents the double scaling limit of quartic tensor models. In the first part of this chapter it is shown show that the melonic family in the quartic melonic tensor model can be analytically resumed and that this resummation is encoded in a translation of the intermediate matrix fields to a nontrivial vacuum. The most striking consequence of this translation is that one can now show that the continuum limit of tensor models corresponds to a genuine phase transition (in the field theory sense) associated to a symmetry breaking. In the second part of this chapter the intermediate field representation is used to derive the double scaling limit of the quartic melonic model.

•Oct 6, 2016•0 citations
The definition of a double-scaling limit represents an important goal in the development of tensor models.We take the first steps towards this goal by extracting and analysing the next-to-leading order contributions, in the 1/N expansion, for the iid tensor models.We show that the radius of convergence of the nlo series coincides with that of the leading order melonic sector.Meanwhile, the value of the susceptibility exponent, γnlo = 3/2, signals a departure from the leading order behaviour.Both pieces of information provide clues for a non-trivial double-scaling limit, for which we put forward some precise conjecture.
The definition of a double scaling limit represents an important goal in the development of tensor models. We take the first steps towards this goal by extracting and analysing the next-to-leading order contributions, in the 1/N expansion, for the colored tensor models. We show that the radius of convergence of the nlo series coincides with that of the leading order melonic sector. Meanwhile, the value of the susceptibility exponent, γ nlo = 3/2, signals a departure from the leading order behavior. Both pieces of information provide clues for a non-trivial double scaling limit, for which we put forward some precise conjecture.
In this paper we study the double scaling limit of the multi-orientable tensor model. We prove that, contrary to the case of matrix models but similarly to the case of invariant tensor models, the double scaling series are convergent. We resum the double scaling series of the two-point function and of the leading singular part of the four-point function. We discuss the behavior of the leading singular part of arbitrary correlation functions. We show that the contribution of the four-point function and of all the higher point functions are enhanced in the double scaling limit. We finally show that all the correlation functions exhibit a singularity at the same critical value of the double scaling parameter which, combined with the convergence of the double scaling series, suggest the existence of a triple scaling limit.
Another proof of the 1/N1/N1/N expansion of the rank three tensor model with tetrahedral interaction
The rank three tensor model with tetrahedral interaction was shown by Carrozza and Tanasa to admit a 1/N1/N1/N expansion, dominated by melonic diagrams, and double tadpoles decorated with melons at next-to-leading order. This model has generated a renewed interest in tensor models because it has the same large NNN limit as the SYK model. In contrast with matrix models, there is no method which would be able to prove the existence of 1/N1/N1/N expansions in arbitrary tensor models. The method used by Carrozza and Tanasa proves the existence of the 1/N1/N1/N expansion using two-dimensional topology, before identifying the leading order and next-to-leading graphs. However, another method was required for complex, rank three tensor models with planar interactions, which is based on flips. The latter are moves which cut two propagators of Feynman graphs and reglue them differently. They allow transforming graphs while tracking their orders in the 1/N1/N1/N expansion. Here we use this method to re-prove the results of Carrozza and Tanasa, thereby proving the existence of the 1/N1/N1/N expansion, the melonic dominance at leading order and the melon-decorated double tadpoles at next-to-leading order, all in one go.
Abstract In the first section we give a briefly presentation of the U(N)D-invariant tensor models (N being again the size of the tensor, and D being the dimension). The next section is then dedicated to the analysis of the Dyson–Schwinger equations (DSE) in the large N limit. These results are essential to implement the double scaling limit mechanism of the DSEs, which is done in the third section. The main result of this chapter is the doubly-scaled 2-point function for a model with generic melonic interactions. However, several assumptions on the large N scaling of cumulants are made along the way. They are proved using various combinatorial methods.
Tensor models and tensor field theories admit a 1/N expansion and a melonic large N limit which is simpler than the planar limit of random matrices and richer than the large N limit of vector models. They provide examples of analytically tractable but non-trivial strongly coupled quantum field theories and lead to a new class of conformal field theories. We present a compact introduction to the topic, covering both some of the classical results in the field, like the details of the 1/N expansion, as well as recent developments. These notes are loosely bases on four lectures given at the Journées de physique mathématique Lyon 2019: Random tensors and SYK models.
Abstract This chapter presents the double scaling limit of quartic tensor models. In the first part of this chapter it is shown show that the melonic family in the quartic melonic tensor model can be analytically resumed and that this resummation is encoded in a translation of the intermediate matrix fields to a nontrivial vacuum. The most striking consequence of this translation is that one can now show that the continuum limit of tensor models corresponds to a genuine phase transition (in the field theory sense) associated to a symmetry breaking. In the second part of this chapter the intermediate field representation is used to derive the double scaling limit of the quartic melonic model.